A steady-state, one-phase, Stefan problem corresponding to the solidification process of an ingot of pure metal by continuous casting with nonlinear lateral cooling is considered via the weak formulation introduced by H. Brezis et al . [C. R. Acad. Sci. Paris Ser. A 287 (1978), no. 9, 711--714] for the dam problem. Two existence results are obtained, for a general nonlinear flux and for a maximal monotone flux. Comparison results and the regularity of the free boundary are discussed. A uniqueness theorem is given for the monotone case.